Old Challenge. In the September ACBL Bridge Bulletin, Noreen Wurdemann of the Bahamas reports that the two bridge hands below were dealt out at her club. She says that she is "100% certain there was no hanky-panky." What is the probability of this happening among say 28 boards? somewhere in the world in a year? in a century? How do you explain this report?

Answer (Joseph DeVincentis). "This pair of hands has so many coincidences about it that, if there was indeed "no hanky-panky", it is likely to be the unlikeliest pair of bridge hands ever.

First off, the hands are identical, except that the east and west hands are swapped. The number of distinct bridge deals, with the hands in the same order, is 52!/13!4 = about 5 x 1028, and so the likelihood of two specific hands being identical is 1 in that many deals.

If you allow any rearrangement of the hands, divide the above number by 4! = 24 to get 2 x 1027.

Among 28 deals there are 378 pairs of deals, and the likelihood of any two of them having the same hands is about 1 in 6 x 1024.

For the world, assume that 1% of the world population (60 million) plays bridge regularly (probably a vast overestimate), and assume that all the bridge played in the world can be accounted for as 20 hands a week or 1000 hands a year for each of these people. That's 60 billion hands a year, or 15 billion deals a year. This gives 1020 pairs of deals, so it's starting to become barely imaginable for the same deal to happen twice -- but by considering all the deals in the world, nobody will ever notice when it happens, since they will likely be hundreds or even thousands of miles apart, in different parts of the year, and with none of the same players involved. If we extend this to all the deals in a century, it starts to become likely that the same hand is dealt twice somewhere, sometime, and even less likely that it will be noticed when it happens.

What I would expect is most likely in this case is that the cards that were dealt to form these hands were taken from matching boards left over from some large duplicate event where identical boards were intentionally made, and both dealers failed to shuffle [probably at the same table, since the Boards are numbered 21 and 24, as observed by Bob Swanson], and either the dealers dealt cards in opposite directions, or the hands got mixed around after dealing. (Or, some worse hanky-panky.)

However, what is even more astounding is that the individual hands mirror each other as perfectly as is possible. Each player holds the same cards as his partner, but with the major suits swapped and the minor suits swapped. The probability of one deal turning out this way is quite low. First dealing, say, 13 cards to the north hand, there are 52 ways to choose the first card, but then the counterpart to that card is forced to go into the south hand, so there are only 50 possibilities for the second card, and so on, moving on to the west hand and having counterparts forced into the east hand for the second half of the deck. This gives an "even factorial" of 52 x 50 x 48 x ... x 6 x 4 x 2 which equals 226 x 26!. Then we divide by 13!2 since we can deal the cards to the north and west hands in any order, to get 226 x 26!/13!2 = about 7 x 1014. The chances of such a deal (one deal!) are thus 1 in 7 x 1013. The chances of ANY two such deals coming up in such a small sample (28 boards) are thus of the same magnitude as any two identical deals coming up in the same small sample.

To have both of these happen together is far less likely -- essentially the probability on the order of 10-24 of two like boards being dealt in the same club on the same night, multiplied by the probability that the duplicated hand is such an unusual one as this, makes it be on the order of 1 in 1038, and still unlikely when you consider all the bridge played in a century."

Swanson adds: "One small further observation that adds a bit of support to DeVincentis' conjecture. While at my local duplicate club this week I checked where boards 21-24 begin the evening in a 28-board layout. Given the details of how the boards circulate, boards 21-24 are not among those that are initially shuffled at the start of a game. They typically sit on a separate table or counter where they would be shuffled if somebody had an extra moment or by the first people to use them. Through inattention, sometimes (not often) they enter play without being shuffled."

New Challenge. My colleague Dick De Veaux reports that upon returning from a three-week trip, he found his checking account balance at exactly $0.00. Without further information, how would you estimate the probability of such an event?

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